The asymptotic relation (REF ) deals with the quotient \({\mathbb {E}}\,f_k(D_N)/\binom{N}{k}\) . Instead of this, we can also consider the difference \(\binom{N}{k}- {\mathbb {E}}\,f_k(D_N)\) . Here a smaller critical number appears, which first showed up in related work of Donoho and Tanner [1]}. To recall it, we denote by \({\sf H}\)  the binary entropy function with base \(e\) ,
\( {\sf H}(x):= -x\log x-(1-x)\log (1-x)\quad \mbox{for } 0\le x\le 1,\) 
